Nonlocal Boundary Conditions Research Articles

Solving systems of coupled nonlinear Atangana–Baleanu-type fractional differential equations

In this work, we investigate two types of boundary value problems for a system of coupled Atangana–Baleanu-type fractional differential equations with nonlocal boundary conditions. The fractional derivatives are applied to serve as a nonlocal and nonsingular kernel. The existence and uniqueness of solutions for proposed problems using Krasnoselskii’s and Banach’s fixed-point approaches are established. Moreover, nonlinear analysis is used to build the Ulam–Hyers stability theory. Subsequently, we discuss two compelling examples to demonstrate the utility of our study.

Nonzero positive solutions of fractional Laplacian systems with functional terms

AbstractWe study the existence of nonzero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with nonlocal functional weights and functional boundary conditions. We present two examples to shed light on the type of functionals and growth conditions that can be considered with our approach.

Existence of multiple solutions for a wide class of differential inclusions†

AbstractWe give a unified approach to study the existence of multiple positive solutions of nonlinear differential inclusions of the form subject to various nonlocal boundary conditions. We study these problems via a perturbed integral inclusion of the form .

Asymptotic analysis of Sturm-Liouville problem with Robin and two-point boundary conditions

We analyze the initial value problem and get asymptotic expansions for solution. We investigate the characteristic equation for Sturm-Liouville problem with one classical Robin type boundary condition and another two-point nonlocal boundary condition. Finally, we obtain asymptotic expansions for eigenvalues and eigenfunctions.

Applicability of Mönch’s Fixed Point Theorem on a System of (k, ψ)-Hilfer Type Fractional Differential Equations

In this article, we study a system of Hilfer (k,ψ)-fractional differential equations, subject to nonlocal boundary conditions involving Hilfer (k,ψ)-derivatives and (k,ψ)-integrals. The results for the mentioned system are established by using Mönch’s fixed point theorem, then the Ulam–Hyers technique is used to verify the stability of the solution for the proposed system. In general, symmetry and fractional differential equations are related to each other. When a generalized Hilfer fractional derivative is modified, asymmetric results are obtained. This study concludes with an applied example illustrating the existence results obtained by Mönch’s theorem.

Solvability of mixed problems for heat equations with two nonlocal conditions

Abstract In this study, the solvability of a problem of the heat conduction theory with two nonlocal boundary conditions is investigated. Systems of eigenfunctions of the corresponding operator with two nonlocal boundary conditions are taken into consideration. A theorem on the solvability of the problem of the theory of heat conduction with two nonlocal boundary conditions is given

Inverse nodal problems for Sturm-Liouville equation with nonlocal boundary conditions

In this paper, a Sturm–Liouville problem with some nonlocal boundary conditions of the Bitsadze-Samarskii type is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of the potential function and some other coefficients in the boundary conditions.

Existence of solutions for fractional differential equations with Ψ-Hilfer fractional derivative with nonlocal integral boundary conditions

Existence of solutions for fractional differential equations with Ψ-Hilfer fractional derivative with nonlocal integral boundary conditions

Numerical Determination of Thermal Conductivity in Heat Equation under Nonlocal Boundary Conditions and Integral as Over specified Condition

In this article, an inverse problem of finding timewise-dependent thermal conductivity has been investigated numerically. Numerical solution of forward (direct) problem has been solved by finite-difference method (FDM). Whilst, the inverse (indirect) problem solved iteratively using Lsqnonlin routine from MATLAB. Initial guess for unknown coefficient expressed by explicit relation based on nonlocal overdetermination conditions and intial input data .The obtained numrical results are presented and discussed in several figures and tables. These results are accurate and stable even in the presense of noisy data.

On periodic solutions of linear parabolic problems with nonlocal boundary conditions

A linear parabolic equation with nonlocal boundary conditions of the BitsadzeSamarsky type is considered. The existence and uniqueness theorem of the periodic solution is proved.

An inverse boundary value problem for transverse vibrations of a bar

In this article, we study an inverse problem (IP) for a fourth-order hyperbolic equation with nonlocal boundary conditions. This IP is reduced to the not self-adjoint boundary value problem (BVP) with corresponding boundary condition. Then, we use the separation of variables method, to reduce the not self-adjoint BVP to an integral equation. The existence and uniqueness of the integral equation are established by the contraction mappings principle and it is concluded that this solution is unique for a not-adjoint BVP. The existence and uniqueness of a nonlocal BVP with integral condition is proved. In addition, the fourth-order hyperbolic PDE is discretized using a collocation technique based on the quintic B-spline (QnB-spline) functions and reformed by the Tikhonov regularization function. The noise and analytical data are considered. The numerical outcome for a standard numerical example is discussed. Furthermore, the stability of the discretized system is also analyzed. The rate of convergence (ROC) of the method is also obtained.

Numerical solution of two-dimensional inverse time-fractional diffusion problem with non-local boundary condition using a-polynomials

Numerical solution of two-dimensional inverse time-fractional diffusion problem with non-local boundary condition using a-polynomials

A LYAPUNOV-TYPE INEQUALITY FOR A HIGHER ORDER FRACTIONAL BOUNDARY VALUE PROBLEM

In this work, we obtain a Lyapunov-type inequality for a\ fractional differential equation involving Caputo fractional derivatives of higher order and subject to nonlocal boundary conditions. An application to eigenvalue problems is also given.

Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions

Abstract In this paper, we analyses the existence and Hyers–Ulam stability of a coupled system of three sequential fractional differential equations with coupled integral boundary conditions. This manuscript can be categorized into three parts: The Leray–Schauder alternative is used to prove the existence of a solution in the first section. The second section emphasizes the analysis of uniqueness, which is based on the Banach fixed point theorem’s concept of contraction mapping, and the third section establishes the Hyers–Ulam stability results. In addition, we provide examples to demonstrate our findings.

The Oberbeck–Boussinesq system with non-local boundary conditions

We consider the Oberbeck–Boussinesq system with non-local boundary conditions arising as a singular limit of the full Navier–Stokes–Fourier system in the regime of low Mach and low Froude numbers. The existence of strong solutions is shown on a maximal time interval [ 0 , T m a x ) [0, T_{\mathrm {max}}) . Moreover, T m a x = ∞ T_{\mathrm {max}} = \infty in the two-dimensional setting.

Numerical treatment of singularly perturbed parabolic partial differential equations with nonlocal boundary condition

This paper presents numerical treatments for a class of singularly perturbed parabolic partial differential equations with nonlocal boundary conditions. The problem has strong boundary layers atx= 0 andx= 1. The nonstandard finite difference method was developed to solve the considered problem in the spatial direction, and the implicit Euler method was proposed to solve the resulting system of IVPs in the temporal direction. The nonlocal boundary condition is approximated by Simpsons13rule. The stability and uniform convergence analysis of the scheme are studied. The developed scheme is second-order uniformly convergent in the spatial direction and first-order in the temporal direction. Two test examples are carried out to validate the applicability of the developed numerical scheme. The obtained numerical results reflect the theoretical estimate.

Free vibration analysis of graded porous circular micro/nanoplates with various boundary conditions based on the nonlocal elasticity theory

AbstractIn this article, the free vibrations of graded porous circular nanoplates subjected to various boundary conditions have been investigated. It is assumed that the performance of graded porous materials varies continuously in the whole thickness, and two cosine forms of non‐uniform porosity distribution along its thickness are considered. In order to capture the size effect, the Eringen's nonlocal elastic theory is applied. Then, Mindlin plate theory, combined with Hamilton's principle, is utilized to derive the governing equations of motion. Various types of boundary conditions are assumed for the plate with a combination of clamped, simply supported and free edges. Finally, the governing equations of motion are solved numerically using the shooting technique. The effects of various factors such as porosity coefficient, porosity distribution pattern, thickness‐diameter ratio, and the nonlocal scale effect and boundary conditions on the natural frequencies of grade porous circular nanoplates are discussed in detail.

Existence and Hyers–Ulam stability of solutions to a nonlinear implicit coupled system of fractional order

Abstract In this typescript, we study system of nonlinear implicit coupled differential equations of arbitrary (non–integer) order having nonlocal boundary conditions on closed interval [0, 1] with Caputo fractional derivative. We establish sufficient conditions for the existence, at least one and a unique solution of the proposed coupled system with the help of Krasnoselskii’s fixed point theorem and Banach contraction principle. Moreover, we scrutinize the Hyers–Ulam stability for the considered problem. We present examples to illustrate our main results.

Systems of Riemann–Liouville Fractional Differential Equations with ρ-Laplacian Operators and Nonlocal Coupled Boundary Conditions

In this paper, we study the existence of positive solutions for a system of fractional differential equations with ρ-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.

Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρk,ϕk)-Hilfer Fractional Integro-Differential Equations

In this paper, we establish the existence and stability results for the (ρk,ϕk)-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the (ρk,ϕk)-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results.